His dad’s quips have resulted in more than 231,000 followers under the account name @s—mydadsays. But after it attracted wide attention in recent weeks as followers retweeted postings, and blogs and mainstream media covered it, Mr. Halpern finally broke the news.

But his dad didn’t really get it. “He asked if I had to start up the Internet to get on Twitter, and then I felt better because I knew he wouldn’t understand the grand scope of what was going on,” said Mr. Halpern in an interview.

Shortly after the revelation, his dad, who doesn’t have Internet on his personal computer by choice, changed the subject…